Systems and Methods for Shaped Single Carrier Orthogonal Frequency Division Multiplexing with Low Peak to Average Power Ratio

ABSTRACT

System and methods for shaped single carrier orthogonal frequency division multiplexing with low peak to average power ratio are provided. The system receives an input signal and modulates the input signal to form Dirichlet kernels in a time domain to generate an offset Dirichlet kernel output time array where each Dirichlet kernel has a main lobe and a plurality of side lobes. Modulating the input signal suppresses a peak to average power ratio of the offset Dirichlet kernel output time array by reducing the plurality of side lobes of each Dirichlet kernel and respective amplitudes of the side lobes.

RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationSer. No. 63/014,682 filed on Apr. 23, 2020 and U.S. Provisional PatentApplication Ser. No. 63/129,956 filed on Dec. 23, 2020, each of which ishereby expressly incorporated by reference.

BACKGROUND Technical Field

The present disclosure relates generally to the field of signalprocessing. More particularly, the present disclosure relates to systemsand methods for shaped single carrier orthogonal frequency divisionmultiplexing with low peak to average power ratio.

Related Art

Orthogonal Frequency Division Multiplexing (OFDM) signals have beenapplied to communication systems operating in frequency-selective fadingchannels due in part to their equalizer structure. OFDM signals cansuffer from large peak to average power ratio (PAPR). This requiressignificant back-off of an average power level in a power amplifier (PA)to avoid clipping and the associated spectral re-growth due to theclipping. The PA back-off decreases an efficiency of the PA andavailable power from a transmitter.

Advanced OFDM-based systems can utilize a combination of techniques tomitigate excessive PAPR. For example, in a cellular system (e.g., 3GPP,4G, or 5G), a modified waveform known as a Single-Carrier OFDM (SC-OFDM)is utilized in a mobile uplink segment where PA inefficiency cannegatively impact battery life. The tradeoff for power efficiency inSC-OFDM is that spectral efficiency is approximately 50% or lesscompared with an unmodified OFDM utilized in a mobile downlink segment.Distortion minimizing techniques can be applied to the PA (e.g., dynamicenvelop-tracking bias adjustment or complex pre-distortion) whichrequire added real-time adjustment to either the PA bias or the modemoutput as a function of the instantaneous amplitude at an input of thePA. These PA compensation techniques can require additionalcomputational and hardware resources which can increase a cost of userequipment.

Thus, what would be desirable is a system that automatically andefficiently processes signals using a SC-OFDM modulator with low peak toaverage power ratio. Accordingly, the systems and methods disclosedherein solve these and other needs.

SUMMARY

This present disclosure relates to systems and methods for shaped singlecarrier orthogonal frequency division multiplexing with low peak toaverage power ratio. The system receives an input signal and modulatesthe input signal to form Dirichlet kernels in a time domain to generatean offset Dirichlet kernel output time array where each Dirichlet kernelhas a main lobe and a plurality of side lobes. Modulating the inputsignal suppresses a peak to average power ratio of the offset Dirichletkernel output time array by reducing the plurality of side lobes of eachDirichlet kernel and respective amplitudes of the side lobes. The systemmodulates the input signal by receiving the input signal by an N-pointtime input array and transforming the N-point time input array to thefrequency domain by a discrete Fourier transform to generate an N-pointinput frequency array. The system replicates the N-point input frequencyarray to generate an M-point input frequency array where M is greaterthan N and utilizes a filter to generate a shaped M-point outputfiltered frequency array by multiplying the M-point input frequencyarray and the filter. The system transforms the shaped M-point outputfiltered frequency array by an inverse discrete Fourier transform togenerate an M-point offset Dirichlet kernel output time array. Thesystem generates a cyclic prefix time array by replicating durationpoints of an end of the M-point offset Dirichlet kernel output timearray, and appends the cyclic prefix time array to a beginning of theM-point offset Dirichlet kernel output time array to generate an M-pointand duration point output time array.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing features of the invention will be apparent from thefollowing Detailed Description of the Invention, taken in connectionwith the accompanying drawings, in which:

FIG. 1A is a graph illustrating a sampled data rectangle envelope h(n);

FIG. 1B is a graph illustrating a sampled data Fourier transform H(θ) ofFIG. 1A;

FIG. 1C is a graph illustrating a discrete Fourier transform H(k) ofFIG. 1A;

FIGS. 2A-2B are graphs illustrating the respective magnitudes of twoOFDM signal sets;

FIGS. 3A-3B are magnitude histograms obtained from 1000 symbols of thetwo OFDM signal sets of FIGS. 2A-B;

FIG. 3C is a graph illustrating complementary cumulative densityfunctions of the OFDM signal sets of FIGS. 3A-3B;

FIG. 4 is a diagram illustrating a SC-OFDM signal generator for formingoffset Dirichlet kernels in the time domain;

FIG. 5A is a graph illustrating a spectrum of a 64-point Dirichletkernel in a 256-point FFT;

FIGS. 5B-5C are graphs respectively illustrating a time impulse responseof the signal generator of FIG. 4 and a phase shift of the time impulseresponse of the signal generator of FIG. 4;

FIGS. 6A and 6B are magnitude histograms obtained from 1000 symbols of16-QAM OFDM and 16-QAM SC-OFDM symbol sets;

FIG. 7 is a graph illustrating complementary cumulative densityfunctions of the 16-QAM OFDM and 16-QAM SC-OFDM symbol sets of FIGS.6A-6B;

FIG. 8 is a diagram illustrating an embodiment of the system of thepresent disclosure;

FIGS. 9A-9B are diagrams respectively illustrating a shaped SC-OFDMmodulator and a shaped SC-OFDM demodulator of the system of FIG. 8;

FIG. 10 is a flowchart illustrating overall processing steps carried outby the shaped SC-OFDM modulator of FIG. 9A;

FIG. 11 is a flowchart illustrating overall processing steps carried outby the shaped SC-OFDM demodulator of FIG. 9B;

FIG. 12 a graph illustrating a frequency response of a shaped SC-OFDMfilter;

FIG. 13 is a series of graphs illustrating respective impulse responsesof a shaped SC-OFDM modulator as a window length of an embedded shapedSC-OFDM filter is shortened;

FIG. 14 is a series of graphs illustrating impulse responses ofrespective shaped SC-OFDM filters;

FIGS. 15A-15D are graphs illustrating complementary cumulative densityfunctions of an OFDM modulator, an SC-OFDM modulator, and a shapedSC-OFDM modulator for a range of excess bandwidths;

FIGS. 16A-16B are graphs illustrating a magnitude time series of ashaped SC-OFDM demodulator for specified excess bandwidths; and

FIG. 17 is a diagram illustrating another embodiment of the system ofthe present disclosure.

DETAILED DESCRIPTION

The present disclosure relates to systems and methods for shaped singlecarrier orthogonal frequency division multiplexing with low peak toaverage power ratio, as described in detail below in connection withFIGS. 1-17.

By way of background, Orthogonal Frequency Division Multiplexing (OFDM)utilizes the Inverse Discrete Fourier Transform (IDFT) as a modulatorand a Discrete Fourier Transform (DFT) as a demodulator. Generally, theIDFT and the DFT are implemented by one of an Inverse Forward FastFourier Transform (IFFT) or a Forward Fast Fourier Transform (FFT)algorithms. It should be understood that the baseband representation ofthe modulated signal is a weighted sum of complex sinusoids. The complexsinusoids are the basis sequences of the DFT process where the sinusoidsspan N samples and include an integer number of cycles per N samples.These sinusoidal sequences are mutually orthogonal. The orthogonalitycan be understood as the inner product of two complex sinusoids thatresults in a sum equal to zero when the sinusoid frequencies aredifferent or a sum equal to N when the sinusoid frequencies are thesame.

In the OFDM process, an amplitude of the complex sinusoid is constantand is equivalent to a scaled rectangle window or gating sequence. Thescaling term is formed by a mapping process from input bit sequences toan amplitude of cosine and sine at each frequency. It should beunderstood that a sampled complex sinusoid forms a continuous periodicspectrum which can be observed by a sampled DFT. A shape of the spectrumis a Dirichlet kernel which is a periodically extended version of thesin(x)/(x) or sinc function as shown below by Equation 1:

$\begin{matrix}{{h(n)} = \{ {{{\begin{matrix}{1:} & {{{- ( {N - 1} )}/2} \leq {{8\%} + {( {N - 1} )/2}}} \\{0:} & {elsewhere}\end{matrix}{H(\theta)}} = \frac{\sin( {N\frac{\theta}{2}} )}{\sin( \frac{\theta}{2} )}},{{- \pi} \leq \theta < \pi}} } & {{Equation}\mspace{14mu} 1}\end{matrix}$

Equation 1 describes a non-causal version of the time series h(n). Theactual time series h(n) would be causal with non-zero samples located inthe interval 0≤n≤N−1 and the spectrum H(θ) containing a phase shift termreflecting a time delay of the causal sequence version of h(n).

The DFT forms H(k) which are uniformly spaced samples of the continuousspectrum H(θ) where θ=k 2π/N. These sample locations correspond to thefrequencies of the complex sinusoids with integer cycles per intervaland coincide with the zeros of the spectrum H(θ). It should beunderstood that the zero locations are also the zeros of the Z-transformH(Z) of the sequence h(n). FIGS. 1A-1C, respectively, illustrate thesequence h(n), the function H(θ), and the sequence H(k). In particular,FIG. 1A illustrates a sampled data rectangle envelop h(n), FIG. 1Billustrates the sampled data Fourier transform H(θ) of h(n) and FIG. 1Cillustrates the DFT H(k) of h(n).

The initial OFDM sampled data sequence formed by the IDFT is shown byEquation 2 below:

$\begin{matrix}{{h(n)} = {\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}{{H(k)}{\exp( {j\frac{2\pi}{N}{nk}} )}}}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

The amplitudes H(k) are complex numbers corresponding to constellationsample points of the M In-Phase and Quadrature grid values of theselected constellation suite M Quadrature Amplitude Modulated (M-QAM).Typical values include, but are not limited to, 64-QAM and 256-QAM.

A subset of the spectral weights H(k) have zero values. The zero valuedweights are utilized to form an empty spectral span between the periodicspectral replicates of the occupied frequency band associated with thesampled data Fourier transform. This empty spectral interval is requiredby the process that utilizes Digital to Analog Converters (DACs) andsmoothing filters to form continuous analog waveforms from the sampleddata sequences. The empty spectral interval permits reasonabletransition bandwidth requirements of the analog filters following theDAC. An interpolator following the output of the IDFT can be utilized toraise the sample rate of the modulated sampled data sequence. Theresulting higher sample rate increases a spacing between the spectralreplicates and reduces the order and therefore a cost of the analogfilters following the DAC. The IDFT can also be utilized to interpolatethe output time series while performing the modulation process. The IDFTcan raise the sample rate and thereby increase the spacing betweenspectral replicates by performing a larger IDFT (e.g., by a factor of 2)and placing additional zero valued spectral samples in the increasednumber of spectral bins.

The output time sequence formed by Equation 2 will now be described. Asshown in Equation 2, the amplitudes H(k) can be a list of random numbersdrawn from a list of possible amplitudes. The multiple weighted sums ateach position “n” in h(n) is then a sum of a large number of identicallydistributed random variables. By the central limit theorem, for eachsample position, an amplitude of the in phase and an amplitude of thequadrature phase time is a Gaussian distributed random variable and theamplitude of the resulting complex number at each sample position isRayleigh distributed. The Rayleigh distribution exhibits long tails withpeak excursions between 3 to 4 times the root mean square (rms) signallevel. As such, a final output power amplifier (PA) would have to backoff approximately 9.5 to 12 dB to avoid clipping of the peak excursionsignal levels. This is indicative of the high peak to average powerratio (PAPR) problem associated with an OFDM signal set.

For example, FIGS. 2A-2B are graphs 50, 52 illustrating the respectivemagnitudes of two OFDM signal sets. In particular, FIGS. 2A and 2B aregraphs 50, 52, respectively, illustrating Quadrature Phase Shift Keying(QPSK) and 256-QAM modulation with 512 frequency bins in a 1024 pointIFFT where the sampled time signal extends over 1000 OFDM symbolsincluding 1024 samples per symbol. Each signal set is normalized tounity variance and the PAPR of the two signal sets are 3.4 and 3.7respectively. FIGS. 3A-3B are magnitude histograms 60, 62 obtained from1000 symbols of the OFDM signal sets of FIGS. 2A-2B and FIG. 3C is agraph 70 illustrating complementary cumulative density functions of theOFDM signal sets of FIGS. 3A-3B. As shown in FIGS. 3A-3C, a probabilitythat the magnitude exceeds 3 times the average magnitude for a unityvariance signal is approximately 1 in 1000 such that approximately 1 inevery 10 symbols will saturate if the saturation level is 3 times theaverage level. Saturation in OFDM is equivalent to an impulse added tothe time series where the impulse has a broad flat bandwidth whichdisturbs every FFT bin in the symbol.

The high PAPR associated with an OFDM signal set is tolerable becauseOFDM provides for a desirable signal denoted by a collection ofdifferent frequency sinusoids. Sinusoids are special functions which, intheir analog form, when propagating through systems described as linearequations, do not change their shape. Similarly, sinusoids are sequenceswhich, in their sampled data form, when propagating through systemsdescribed as linear difference equations, do not change their shape.Accordingly, sinusoids are eigen-functions (or eigen-sequences) oflinear differential (or difference) systems. Therefore, if a sinewave isdifferentiated and the derivative is scaled and added back to theoriginal sinewave, the resulting sinewave has the same steady stateshape as the original. As such, a system can change a size of a sinewavebut cannot change its shape. Thus, when a sinusoid propagates through achannel and experiences the summation of a delayed and scaled version ofitself due to multipath, the shape is preserved and the effect of thechannel is a complex gain change, magnitude and phase or A(f)xexp(jθ(f)). Equalization, the inversion of the channel when sinusoidspropagate through the channel, becomes a task of estimating thechannel's sinusoidal steady state gain and applying a correctivemultiplicative inverse. Since the modulator and demodulator of the OFDMsignal set is a Fourier transform, the channel sinusoidal steady stategain at each frequency can be determined by probing the channel with apreamble and then applying the channel inversion gain correction topayload symbols with information gleaned from the channel by the probingpreamble. OFDM is advantageous because the channel can be triviallyinverted. This is an important consideration when evaluating widebandwidth channels.

The standard model of a multipath channel is a tapped delay line andfinite impulse response (FIR) filter where its delay spread function isthe impulse response of the filter. Each frequency in a modulated OFDMsignal will have a different impulse response and hence a differentsteady state frequency response. Each frequency in the modulationprocess is probed with a sinusoid at that frequency where the amplitudeand phase at the modulator is known to the demodulator. The ratio of theoutput gain and phase to the input gain and phase presents, to thedemodulator, estimates of the channel gain at each frequency.

During the modulation process it is important to avoid coupling betweensuccessive symbols known as inter symbol interference (ISI). ISI is theresult of channel memory in its frequency dependent point spreadfunction. The coupling of successive symbols can be avoided by insertinga time gap or guard interval between successive symbols with the gapduration exceeding the largest delay spread interval thereby renderingthe ISI equivalent to zero.

Convolution can be problematic during the modulation process. Forexample, convolution of the modulated signal with the channel impulseresponse can cause a starting transient and a stopping transient as thetime signal enters and leaves the channel. This transient causesamplitude modulation of the sinusoids in the signal set which breaks theorthogonality of the original rectangle enveloped signal set.Orthogonality is integral to the modulation process because it isessential to the orthogonal steady state gains of the equalizer process.Therefore, to preserve signal orthogonality, a segment of the back endof the symbol can be copied and appended to a front end of the signal.This appended segment is known as a cyclic prefix. Since the sinewavesin the modulation process include an integer number of cycles, thesummed sequence is periodic in its own length such that the next sampleat the right most edge is the first sample at the left most edge.Copying a segment of the back end to the front edge provides forforming, at the boundary, the periodic extension of the sequence andshifting the discontinuity at the former boundary to the left most edgeof the appended segment. It should be understood that the appendedcyclic prefix fits in the guard interval inserted between successivesymbols. As such, when the sequence with the appended cyclic prefix isconvolved with the channel, the starting transient of the current symbolresides in the interval including the cyclic prefix and the stoppingtransient from the previous symbol also resides in the cyclic prefixinterval. With both transients in the interval including the cyclicprefix, the demodulator discards the cyclic prefix interval.Accordingly, there is no transient at the previous boundary because ofthe continuity of the cyclic prefix with the symbol. An attribute of thecyclic prefix in the convolution of the channel with the cyclic prefixappended signal is that, even though the convolution is a linearconvolution, the cyclic prefix in that convolution makes the convolutionappear to be a circular convolution. Accordingly, the cyclic prefix canbe utilized to fool a linear convolution to become a circular, transientfree circular convolution. This trick preserves the orthogonality of theIDFT basis set presented to and processed by the demodulator.

As described above, an OFDM signal can be formed with an N-pointtransform that forms an N sample interval containing a summation ofweighted complex sinusoids. Each sinusoid contains an integer number ofcycles. This is a sufficient condition for the sinusoid sequences to bemutually orthogonal. The number of sinusoids is less than the size ofthe IDFT as a technique to leave an empty spectral guard intervalbetween spectral replicates. Successive symbols can be appended withguard intervals and cyclic prefaces to move the symbols throughmultipath channels without corrupting a structure of the modulatedsymbols. After signal acquisition, the guard interval of each isidentified and discarded and the channel effects are removed by spectralgain corrections as signal conditioning during the demodulation process.

In a standard OFDM signal, a rectangle envelope is utilized in the timedomain and a Dirichlet kernel is utilized in the frequency domain. Eachof the orthogonal sinewaves utilizes the same time sequence rectangleenvelope to form its signal component. The spectrum of each time domainsinusoid is the Dirichlet kernel offset in the frequency domain which isan integer multiple of (2π/N) to form H(θ−k×2π/N) where k is a number ofcycles per interval in the time as well as the frequency offset index inthe frequency domain.

PAPR can be reduced by interchanging the two functions such that therectangle envelope sinusoids reside in the frequency domain and theoffset Dirichlet kernels reside in the time domain. This interchange ispossible given the symmetry of the Fourier transform but the algorithmicimplementation thereof requires a slight modification. The initial stepof the interchange involves the formation of Dirichlet kernel samples inthe time domain. In this regard, FIG. 4 is a diagram 90 illustrating aSC-OFDM signal generator for forming offset Dirichlet kernels in thetime domain. A 64-point input time array 92 receives, as an input, 64constellation points with a selected grid density to the 64 addresses ofthe 64-point input time array. Then, a 64-point Discrete FourierTransform (DFT) 94 is executed, resulting in a 64-point input frequencyarray 96. The 64 points are then unwrapped into 65 points with a halfamplitude cut at address 32 and the points placed symmetrically aboutaddress 0 of a 256-point input frequency array 98. A 256-point InverseDFT (IDFT) 100 is executed, resulting in the time domain via a 256-pointoutput time array 102 with a 1 to 4 interpolated sum of scaled andoffset Dirichlet kernels. Next, a subset of the samples of the 256-pointoutput time array 102 is appended to a beginning of a subsequent outputtime array as its cyclic prefix thereby yielding a 320-point output timearray 104. As mentioned above, the cyclic prefix can be appended to thesubsequent output time array because every sum of the output sinusoidsis periodic in the length of the output time array as is everyindividual sinewave in the original OFDM signal set.

FIG. 5A is a graph 120 illustrating a spectrum of a 64-point Dirichletkernel in a 256-point FFT and FIGS. 5B-5C are graphs 122, 124respectively illustrating a time impulse response of the signalgenerator of FIG. 4 and a phase shift of the time impulse response ofthe signal generator of FIG. 4. In particular, graphs 120, 122, 124 ofFIGS. 5A-5C illustrate forming samples of a Dirichlet kernel byutilizing FFT and zero extended IDFT to perform time domaininterpolation. Starting with a 64-point FFT, an impulse can be loaded inan address 0 with zeros in the remaining 63 addresses. There is only oneminimum bandwidth time series that passes through these 64 samples and,as such, it is known that this is the Dirichlet kernel. It should beunderstood that the Dirichlet kernel is not visible because the samplerate has not been raised to view the samples between the zeros. A64-point DFT is executed to obtain a spectrum of the single sample. Thetransform of the single impulse located at index 0 is 64 samples of aconstant unit amplitude spectrum. The 64 samples are positionedsymmetrically at an address 0 of a 256-point spectrum array. Thecontents of address 32are divided in half such that half of the contentsremain in address +32 and the other half of the contents are allocatedto address −32 of the 256-point array. A 256-point IDFT is executed toreturn to the time domain with the time series passing through theinitial 64 time samples but interpolated 1 to 4 to locate three samplesbetween the initial input samples.

Graph 120 of FIG. 5A illustrates the constant levels obtained from theoutput array of the 64-point DFT. As mentioned above, these samples arepositioned symmetrically about address 0 in the 256-point frequencyarray such that the half amplitudes at addresses −32 and +32 areevident. Graph 122 of FIG. 5B illustrates the 64 input sample to the64-point input array to the 64-point DFT. A 256-point IDFT of thespectral array containing the zero extended output of the 64-point DFTis executed and the 256 samples are plotted as an overlay 123 on the256-point time array including the initial single impulse at address 0.As such, graph 122 of FIG. 5B illustrates the 1 to 4 up-sampledDirichlet kernel in the time domain. Graph 124 of FIG. 5C illustrates aphase shift with respect to FIGS. 5A-B. It should be understood thatrepeating the interpolation process described above with a singleimpulse located at address +20 would yield a spectrum at the output ofthe 64-point DFT having the same unit magnitude values riding on acomplex sinusoid with 20 cycles per interval of length 64 where thephase term would reflect the time delay or offset from 0 in the timedomain. For example, executing the unwrapping of the 64 spectral pointsincluding the half amplitude partition of address 32, symmetricallypositioning these samples at address 0 of the 256-point time array, andexecuting the 256-point IDFT, would yield a 1 to 4 up-sampled Dirichletkernel centered at address +20 (the initial position of this impulseresponse test). The 256 samples are plotted as an overlay 125 on the256-point time array including the single impulse at address +20.

The Dirichlet kernels can be positioned in the time domain becauseDirichlet kernels are sin(x)/x like signals where each Dirichlet kernelhas a tall main lobe at its center and low level side lobes positionedaway from the main lobe. When performing the sum of OFDM sinewaves, thesum at each location becomes large because the amplitude is carried bythe complex sinusoid to all sample locations. This is not possible whenperforming the sum of SC-OFDM scaled and time offset Dirichlet kernelsbecause the amplitude is localized and is not distributed to all samplelocations. It should be understood that the main lobe of a Dirichletkernel does not overlay the main lobe of a neighboring Dirichlet kernelbut rather overlays the side lobes of the neighboring Dirichlet kernel.Accordingly, the weighted sum of Dirichlet kernels is dominated by thesingle large peaks of the respective Dirichlet kernels.

FIGS. 6A-6B and 7 validate the expected PAPR reduction of the SC-OFDMsignal set. In particular, FIGS. 6A and 6B are magnitude histograms 130,132 obtained from 1000 symbols of 16-QAM OFDM and 16-QAM SC-OFDM symbolsets where each symbol set includes 64 occupied frequency bins in a256-point transform and FIG. 7 is a graph 140 illustrating complementarycumulative density functions for the 16-QAM OFDM and 16-QAM SC-OFDMsymbol sets of FIGS. 6A-6B. As shown in FIGS. 6A and 6B, the magnitudehistogram 130 of the 16-QAM OFDM symbol set illustrates a standardRaleigh distribution while the magnitude histogram 132 of the 16-QAMSC-OFDM symbol set illustrates 3 peaks associated with three radiicircles of a 16-QAM constellation set where inner and outer circlesinclude 4 constellation points and a middle circle includes 8constellation points. As shown in FIG. 7, the SC-OFDM amplitude spreadis reduced compared to the OFDM amplitude spread. In particular, at 10⁻⁴probability, the SC-OFDM magnitude is approximately 72% of the OFDMmagnitude which represents a 2.8 dB reduction in the PAPR.

As described above, the SC-OFDM process replaces complex sinusoids, thetime domain basis functions of the DFT with Dirichlet kernels and thefrequency domain basis functions of the DFT. The weighted sum of timeshifted Dirichlet kernels has a smaller PAPR than the weighted sum ofoverlapped sinusoids (as shown in FIG. 7) due to their reducedinteraction in the summation process. The time delayed Dirichlet kernelsinteract through the low level side lobes of respective kernels.Generally, this interaction is between a main lobe of a Dirichlet kerneland side lobes of a neighboring Dirichlet kernel because the side lobesdecay inversely with a time offset as shown in the 1/x of the sin(x)/xfunction. The system of the present disclosure can realize additionalPAPR suppression by reducing respective amplitudes of the Dirichletkernel time domain side lobes as well as decreasing their number.

FIG. 8 is a diagram illustrating an embodiment of the system 150 of thepresent disclosure. The system 150 could be embodied as system code 156(e.g., firmware, software, etc.) executed by a processor of aradiofrequency transceiver 152. The transceiver could include, but isnot limited to, a cellular transceiver (e.g., base station or mobiledevice supporting one or more communications protocols such as 3GPP, 4G,5G, etc.), a satellite transceiver (e.g., an earth station or asatellite in space), a wireless networking transceiver (e.g., a WiFibase station or WiFi-enabled device), a short-range (e.g., Bluetooth)transceiver, or any other radiofrequency transceiver. The transceiver152 executes the system code 156, which causes the transceiver 152 togenerate a modified SC-OFDM signal set to reduce PAPR. Optionally, thecode 156 could obtain one or more OFDM signals from a signal setdatabase 154, if desired.

The code 156 (i.e., non-transitory, computer-readable instructions)could be stored on a computer-readable medium and executable by thetransceiver 152 or one or more computer systems. The code 156 couldinclude various custom-written software modules that carry out thesteps/processes discussed herein, and could include, but is not limitedto, a shaped SC-OFDM modulator 158 a and a shaped SC-OFDM demodulator158 b. The code 156 could be programmed using any suitable programminglanguages including, but not limited to, C, C++, C#, Java, Python or anyother suitable language. Additionally, the code 156 could be distributedacross multiple computer systems in communication with each other over acommunications network, and/or stored and executed on a cloud computingplatform and remotely accessed by a computer system in communicationwith the cloud platform. The code 156 could communicate with the signalset database 154, which could be stored on the same transceiver as thecode 156, or on one or more computer systems in communication with thecode 156.

Still further, the system 150 could be embodied as a customized hardwarecomponent such as a field-programmable gate array (“FPGA”),application-specific integrated circuit (“ASIC”), embedded system, orother customized hardware components without departing from the spiritor scope of the present disclosure. It should be understood that FIG. 8is only one potential configuration, and the system 150 of the presentdisclosure can be implemented using a number of differentconfigurations.

FIGS. 9A-9B are diagrams respectively illustrating a shaped SC-OFDMmodulator 158 a and a shaped SC-OFDM demodulator 158 b of the system150. As shown in FIG. 9A, the shaped SC-OFDM modulator 202 a can includean N-point SC-OFDM offset Dirichlet kernel input time array 204 (e.g., a64-point input time array), an N-point FFT 206 (e.g., a 64-point FFT),an N-point input frequency array 208 (e.g., a 64-point input frequencyarray), an M-point input frequency array 210 (e.g., a 256-point inputfrequency array), a filter spectrum array 212 (e.g., a Nyquist filter orDirichlet kernel filter), an M-point filtered output frequency array 214(e.g., a 256-point output frequency array), an M-point FFT 216 (e.g., a256-point FFT), an M-point output time array 218 (e.g., a 256-pointoutput time array), and an M-point output time array 220 (e.g., a320-point output time array) including a cyclic prefix 222. As shown inFIG. 9B, the shaped SC-OFDM demodulator 202 b can include an N-pointinput time array 240 (e.g., a 160-point input time array) including acyclic prefix 242, an N-point input time array 244 (e.g., a 128-pointinput time array), an N-point FFT 246 (e.g., a 128-point FFT), anM-point input frequency array 248 (e.g., a 128-point frequency array), afilter spectrum array 250 (e.g., a Nyquist filter or Dirichlet kernelfilter), a channel spectrum array 252, an M-point filtered outputfrequency array 254 (e.g., a 128-point output frequency array), anM-point output frequency array 256 (e.g., a 64-point output frequencyarray), an M-point FFT 258 (e.g., a 64-point FFT), and an M-point outputtime array 260 (e.g., a 64-point output time array).

FIG. 10 is a flowchart illustrating overall processing steps 280 carriedout by the shaped SC-OFDM modulator 158 a of the system 150 to generateDirichlet kernel samples in the time domain to form an offset Dirichletkernel output time array. Beginning in step 282, the system 150receives, by an N-point input time array, an input signal. In step 284,the system 150 transforms the N-point input time array to the frequencydomain by a N-point DFT (e.g., a 64-point DFT) to generate an N-pointinput frequency array. Then, in step 286, the system 150 replicates theN-point input frequency array with zero filling to generate an M-pointinput frequency array where M is greater than N.

Next, in step 288, the system 150 utilizes a filter to generate a shapedfiltered frequency array by multiplying the M-point input frequencyarray and the filter which is equivalent to convolution in the timedomain. The filter can be a Nyquist filter or a Dirichlet kernel filterhaving a square root of a low side lobe response characteristic. ANyquist filter is utilized in a traditional QAM modem. It should beunderstood that in a QAM modulator, the Nyquist pulse in linearlyconvolved with the constellation samples while in the SC-OFDM modulator,the Dirichlet kernel is circularly convolved with the constellationsamples with the circular convolution performed by a spectral product inthe frequency domain.

The impulse response of the Nyquist filter is a time domain shape thatsupports a given symbol rate with zero ISI while minimizing themodulation bandwidth. The minimum modulation bandwidth is a spectralrectangle with two-sided bandwidth equal to the modulation rate. A knowndisadvantage of the minimum bandwidth Nyquist pulse is that its impulseresponse is infinitely long because its spectrum is discontinuous. Thiscan be addressed by applying a finite width time domain window to theimpulse response. For example, a window can be a rectangle with valuesof 1 in the desired span of the filter length and 0 elsewhere. The timedomain product of the Nyquist impulse response and the applied finiteduration window forms a finite duration impulse response having aspectrum obtained by the convolution of the ideal rectangular spectrumwith the window's spectrum. Nyquist proposed a window whose transformwas a half cycle of a cosine and the resulting filter is referred to asa cosine tapered Nyquist filter. The spectral convolution of the twofunctions widens the modulation bandwidth beyond the minimum rectangularbandwidth.

It should be understood that since the windowed time impulse responseseries does not have a rectangle spectrum, the system 150 splits theshaping process with the square-root spectra by performing half of theshaping at the shaped SC-OFDM modulator 202 a and half of the shaping atthe shaped SC-OFDM demodulator 202 b similar to Nyquist filtered QAMmodems. For the rectangle shaped spectrum, the shaping filter and thematched filter each have rectangle spectra and, as such, their productis also rectangle shaped. With a spectral taper, each of the shapingfilter and the matched filter have square root spectral shapes and, assuch, their product is the cosine tapered Nyquist spectrum. The filtersutilized in conventional QAM modems are referred to as SQRT cosinetapered Nyquist filters. It should be understood that the system 150 canutilize the SQRT cosine tapered Nyquist filters in the shaped SC-OFDM orcan generate a tapered SQRT filter with an optimal window.

Referring back to FIG. 10, in step 290, the system 150 transforms theshaped filtered frequency array to the time domain by an N-point IDFT(e.g., a 256-point IDFT) to generate an M-point output time array. Instep 292, the system 150 generates a cyclic prefix time array byreplicating duration/length points (e.g., Q-points) of an end of theM-point output time array. It should be understood that theduration/length of the cyclic prefix array can depend on the a transientresponse of the system 150 and a worst-case external reflection delay ina given application. In step 294, the system 150 appends the cyclicprefix to a beginning of the M-point time array to generate an M andQ-point output time array.

FIG. 11 is a flowchart illustrating overall processing steps 300 carriedout by the shaped SC-OFDM demodulator 158 b of FIG. 9B of the system150. In step 302, the system 150 removes the appended cyclic prefix timearray to generate an M-point input time array. Then, in step 304, thesystem 150 transforms the M-point time input array to the frequencydomain by an M-point discrete Fourier transform (e.g., a 128-point DFT)to generate a shaped M-point input frequency array. In step 306, thesystem 150 utilizes a matched filter to generate a shaped M-point inputfiltered frequency array by multiplying the shaped M-point inputfrequency array and the matched filter. Then, in step 308, the system150 applies a channel equalizer to the shaped M-point input filteredfrequency array to generate a shaped M-point output filtered andequalized frequency array. In step 310, the system 150 down samples theshaped M-point output filtered and equalized frequency array by applyingan Alias output spectrum to the shaped M-point output filtered andequalized frequency array to generate an N-point output frequency array.In step 312, the system 150 transforms the N-point output frequencyarray by an N-point inverse discrete Fourier transform (e.g., a 64-pointIDFT) to generate an N-point offset Dirichlet kernel output time array.

FIG. 12 a graph 350 illustrating a frequency response of a shapedSC-OFDM filter. In particular, the graph 350 of FIG. 12 illustrates afrequency response of a rectangle spectrum of a standard SC-OFDMinterpolated with a Dirichlet kernel time series and modified bywindowing the Dirichlet kernel and circularly convolving the spectrum ofeach of the Dirichlet kernel time series spectrum and the window. Itshould be understood that the rectangle spectrum is widened by thecircular convolution of the spectrum of each of the Dirichlet kerneltime series and window. The resulting spectrum is square rooted andinverse transformed where the SQRT spectrum for the spectral shaping issplit at the shaped SC-OFDM modulator 158 a and the shaped SC-OFDMdemodulator 158 b. As shown in FIG. 12, the half amplitude −6 dB gainsat the boundaries are 0.707 amplitude indicative of −3 dB gains.

FIG. 13 is a series of graphs 370 a-f illustrating respective impulseresponses of a shaped SC-OFDM modulator as a window length of anembedded shaped SC-OFDM filter is shortened. As shown in FIG. 13, thespectra of windowed Dirichlet kernels becomes shorter with lower levelside lobes as the window length is shortened.

FIG. 14 is a series of graphs 390 a-j illustrating impulse responses ofrespective shaped SC-OFDM filters. In particular, FIG. 14 illustratesdetails of the time domain impulse response main lobe and adjacent sidelobes with marked levels of different SQRT Nyquist filters embedded inthe shaped SC-OFDM modulator 158 a. The amplitudes of the highestadjacent side lobes are indicated by the arrows of each graph 390 a-j.It should be understood that the reduced level of the side lobes and thediminished number of side lobes provide for reduced levels of PAPR inthe modified SC-OFDM modulators.

FIGS. 15A-15D are graphs illustrating complementary cumulative densityfunctions of an OFDM modulator, a SC-OFDM modulator and a shaped SC-OFDMmodulator for a range of excess bandwidths. FIG. 15A is a graph 410illustrating the complementary cumulative density functions of an OFDMmodulator, a SC-OFDM modulator and a shaped SC-OFDM modulator for arange of excess bandwidths for QPSK constellation. FIGS. 15B-15D aregraphs 420, 430, 440 which respectively illustrate the complementarycumulative density functions of an OFDM modulator, a SC-OFDM modulatorand a shaped SC-OFDM modulator for a range of excess bandwidths for16-QAM constellation, 64-QAM constellation and 256-QAM constellation.

FIGS. 16A-16B are graphs illustrating a magnitude time series of ashaped SC-OFDM demodulator for specified excess bandwidths. Inparticular, graph 460 of FIG. 16A illustrates a 256-QAM modulated OFDMand graph 462 of FIG. 16A illustrates a modulation bandwidth spectrum464 a, a shaping filter bandwidth spectrum 464 b and a modulated timeseries power spectrum 464 c of 1000 symbols. As shown in FIG. 16A, peakand average magnitudes of a segment of the magnitude time series areextracted from 1000 symbols and the 40% excess bandwidth shaping filterspectrum 464 b and the power spectrum 464 c are formed from an ensembleof 300 realizations. Additionally, graph 480 of FIG. 16B illustrates aQPSK modulated OFDM and graph 482 of FIG. 16B illustrates a modulationbandwidth spectrum 484 a, a shaping filter bandwidth spectrum 484 b anda modulated time series power spectrum 484 c of 1000 symbols. As shownin FIG. 16B, peak and average magnitudes of a segment of the magnitudetime series are extracted from 1000 symbols and the 20% excess bandwidthshaping filter spectrum 484 b and the power spectrum 484 c are formedfrom an ensemble of 300 realizations.

FIG. 17 is a diagram illustrating another embodiment of the system ofthe present disclosure. In particular, FIG. 17 illustrates additionalcomputer hardware and network components on which the system 500 couldbe implemented. The system 500 can include a plurality of computationservers 502 a-502 n having at least one processor and memory forexecuting the computer instructions and methods described above (whichcould be embodied as system code 156). The system 500 can also include aplurality of signal set storage servers 504 a-504 n for storing signaldata. The computation servers 502 a-502 n and the signal set storageservers 504 a-504 n can communicate over a communication network 506. Ofcourse, the system 500 need not be implemented on multiple devices, andindeed, the system 500 could be implemented on a single computer system(e.g., a personal computer, server, mobile computer, smart phone, etc.)without departing from the spirit or scope of the present disclosure.

Having thus described the system and method in detail, it is to beunderstood that the foregoing description is not intended to limit thespirit or scope thereof. It will be understood that the embodiments ofthe present disclosure described herein are merely exemplary and that aperson skilled in the art can make any variations and modificationwithout departing from the spirit and scope of the disclosure. All suchvariations and modifications, including those discussed above, areintended to be included within the scope of the disclosure. What isdesired to be protected by Letters Patent is set forth in the followingclaims.

What is claimed is:
 1. A system for modulating a signal, comprising: atransceiver; and a processor in communication with the transceiver, theprocessor: receiving an input signal from the transceiver; andmodulating the input signal to form Dirichlet kernels in a time domainto generate an offset Dirichlet kernel output time array, each Dirichletkernel having a main lobe and a plurality of side lobes, whereinmodulating the input signal suppresses a peak to average power ratio ofthe offset Dirichlet kernel output time array by reducing the pluralityof side lobes of each Dirichlet kernel and respective amplitudes of theside lobes.
 2. The system of claim 1, wherein the transceiver is one ofa cellular transceiver, a satellite transceiver, a wireless networkingtransceiver, a short range transceiver, or a radiofrequency transceiver.3. The system of claim 1, wherein the modulation is single carrierorthogonal frequency division multiplexing.
 4. The system of claim 1,wherein the processor modulates the input signal by: receiving the inputsignal by an N-point time input array, transforming the N-point timeinput array to the frequency domain by a discrete Fourier transform togenerate an N-point input frequency array, replicating the N-point inputfrequency array to generate an M-point input frequency array where M isgreater than N, utilizing a filter to generate a shaped M-point outputfiltered frequency array by multiplying the M-point input frequencyarray and the filter, transforming the shaped M-point output filteredfrequency array by an inverse discrete Fourier transform to generate anM-point offset Dirichlet kernel output time array, generating a cyclicprefix time array by replicating duration points of an end of theM-point offset Dirichlet kernel output time array, and appending thecyclic prefix time array to a beginning of the M-point offset Dirichletkernel output time array to generate an M-point and duration pointoutput time array.
 5. The system of claim 4, wherein the replicationinserts a spectral guard interval between spectral replicates of theN-point input frequency array.
 6. The system of claim 4, wherein thefilter is a square root cosine tapered Nyquist filter.
 7. The system ofclaim 4, wherein the processor demodulates the M-point and durationpoint output time array by: removing the appended cyclic prefix timearray to generate an M-point input time array, transforming the M-pointtime input array to the frequency domain by a discrete Fourier transformto generate a shaped M-point input frequency array, utilizing a matchedfilter to generate a shaped M-point input filtered frequency array bymultiplying the shaped M-point input frequency array and the matchedfilter, applying a channel equalizer to the shaped M-point inputfiltered frequency array to generate a shaped M-point output filteredand equalized frequency array, down sampling the shaped M-point outputfiltered and equalized frequency array to generate an N-point outputfrequency array, and transforming the N-point output frequency array byan inverse discrete Fourier transform to generate an N-point offsetDirichlet kernel output time array.
 8. A method for modulating a signalcomprising the steps of: receiving an input signal; and modulating theinput signal to form Dirichlet kernels in a time domain to generate anoffset Dirichlet kernel output time array, each Dirichlet kernel havinga main lobe and a plurality of side lobes, wherein modulating the inputsignal suppresses a peak to average power ratio of the offset Dirichletkernel output time array by reducing the plurality of side lobes of eachDirichlet kernel and respective amplitudes of the side lobes.
 9. Themethod of claim 8, wherein the modulation is single carrier orthogonalfrequency division modulation.
 10. The method of claim 8, wherein thestep of modulating the input signal comprises: receiving the inputsignal by an N-point time input array, transforming the N-point timeinput array to the frequency domain by a discrete Fourier transform togenerate an N-point input frequency array, replicating the N-point inputfrequency array to generate an M-point input frequency array where M isgreater than N, utilizing a filter to generate a shaped M-point outputfiltered frequency array by multiplying the M-point input frequencyarray and the filter, transforming the shaped M-point output filteredfrequency array by an inverse discrete Fourier transform to generate anM-point offset Dirichlet kernel output time array, generating a cyclicprefix time array by replicating duration points of an end of theM-point offset Dirichlet kernel output time array, and appending thecyclic prefix time array to a beginning of the M-point offset Dirichletkernel output time array to generate an M-point and duration pointoutput time array.
 11. The method of claim 10, wherein the step ofreplicating the N-point input frequency array inserts a spectral guardinterval between spectral replicates of the N-point input frequencyarray.
 12. The method of claim 10, wherein the filter is a square rootcosine tapered Nyquist filter.
 13. The method of claim 10, furthercomprising the steps of demodulating the M-point and duration pointoutput time array by: removing the appended cyclic prefix time array togenerate an M-point input time array, transforming the M-point timeinput array to the frequency domain by the discrete Fourier transform togenerate a shaped M-point input frequency array, utilizing a matchedfilter to generate a shaped M-point input filtered frequency array bymultiplying the shaped M-point input frequency array and the matchedfilter, applying a channel equalizer to the shaped M-point inputfiltered frequency array to generate a shaped M-point output filteredand equalized frequency array, down sampling the shaped M-point outputfiltered and equalized frequency array to generate an N-point outputfrequency array, and transforming the N-point output frequency array bythe inverse discrete Fourier transform to generate an N-point offsetDirichlet kernel output time array.
 14. A non-transitory computerreadable medium having instructions stored thereon for modulating asignal which, when executed by a processor of a transceiver, causes theprocessor to carry out the steps of: receiving an input signal; andmodulating the input signal to form Dirichlet kernels in a time domainto generate an offset Dirichlet kernel output time array, each Dirichletkernel having a main lobe and a plurality of side lobes, whereinmodulating the input signal suppresses a peak to average power ratio ofthe offset Dirichlet kernel output time array by reducing the pluralityof side lobes of each Dirichlet kernel and respective amplitudes of theside lobes.
 15. The non-transitory computer readable medium of claim 14,wherein the transceiver is one of a cellular transceiver, a satellitetransceiver, a short range transceiver, a wireless networkingtransceiver, or a radiofrequency transceiver.
 16. The non-transitorycomputer readable medium of claim 14, wherein the modulation is singlecarrier orthogonal frequency division modulation.
 17. The non-transitorycomputer readable medium of claim 14, the processor carrying out thestep of modulating the input signal by: receiving the input signal by anN-point time input array, transforming the N-point time input array tothe frequency domain by a discrete Fourier transform to generate anN-point input frequency array, replicating the N-point input frequencyarray to generate an M-point input frequency array where M is greaterthan N, utilizing a filter to generate a shaped M-point output filteredfrequency array by multiplying the M-point input frequency array and thefilter, transforming the shaped M-point output filtered frequency arrayby an inverse discrete Fourier transform to generate an M-point offsetDirichlet kernel output time array, generating a cyclic prefix timearray by replicating duration points of an end of the M-point offsetDirichlet kernel output time array, and appending the cyclic prefix timearray to a beginning of the M-point offset Dirichlet kernel output timearray to generate an M-point and duration point output time array. 18.The non-transitory computer readable medium of claim 17, wherein thestep of replicating the N-point input frequency array inserts a spectralguard interval between spectral replicates of the N-point inputfrequency array.
 19. The non-transitory computer readable medium ofclaim 17, wherein the filter is a square root cosine tapered Nyquistfilter.
 20. The non-transitory computer readable medium of claim 17, theprocessor further carrying out the step of demodulating the M-point andduration point output time array by: removing the appended cyclic prefixtime array to generate an M-point input time array, transforming theM-point time input array to the frequency domain by the discrete Fouriertransform to generate a shaped M-point input frequency array, utilizinga matched filter to generate a shaped M-point input filtered frequencyarray by multiplying the shaped M-point input frequency array and thematched filter, applying a channel equalizer to the shaped M-point inputfiltered frequency array to generate a shaped M-point output filteredand equalized frequency array, down sampling the shaped M-point outputfiltered and equalized frequency array to generate an N-point outputfrequency array, and transforming the N-point output frequency array bythe inverse discrete Fourier transform to generate an N-point offsetDirichlet kernel output time array.